chap8 gc mit

8/2/2019 Chap8 Gc Mit

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Chapter 8

The general circulation of the

atmosphere

In this chapter we return to our discussion of the large-scale circulation of theatmosphere and make use of the dynamical ideas developed in the previoustwo chapters to enquire into its cause. We begin by reviewing the demandson the atmospheric circulation imposed by global energy and angular mo-mentum budgets as depicted in Fig.8.1. The atmosphere must transportenergy from equator to pole to maintain the observed pole-equator tempera-ture gradient. In addition, because there are westerly winds at the surface inmiddle latitudes and easterly winds in the tropics (see, e.g., Fig.7.28) angu-lar momentum must be transported from the tropics to higher latitudes. Wediscuss how this transfer is achieved by the Hadley circulation in the trop-ics and weather systems in middle latitudes. We will discover that weathersystems arise from a hydrodynamic instability of the thermal wind shearassociated with the pole-equator temperature gradient. Produced througha mechanism known as baroclinic instability, the underlying dynamics areillustrated through laboratory experiment and discussed from the perspec-tive of the available potential energy stored by the horizontal temperaturegradients.

8.1 Understanding the observed circulation

The simplest observed global characteristic of the atmosphere is that thetropics are much warmer than the poles. As discussed in Chapter 5, this

249


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250CHAPTER 8. THE GENERAL CIRCULATION OF THE ATMOSPHERE

Figure 8.1: Latitudinal transport of (left) heat and (right) angular momentumimplied by the observed state of the atmosphere. In the energy budget there is

a net radiative gain in the tropics and a net loss at high latitudes; in order to

balance the energy budget at each latitude, a poleward heat flux is implied. In

the angular momentum budget the atmosphere gains angular momentum in low

latitudes (where the surface winds are easterly) and loses it in middle latitudes

(where the surface winds are westerly). A poleward atmospheric flux of angular

momentum is thus implied.


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8.1. UNDERSTANDING THE OBSERVED CIRCULATION 251

Figure 8.2: Schematic of the observed atmospheric general circulation for annual-averaged conditions. The upper level westerlies are shaded to reveal the core of

the subtropical jet stream on the poleward flank of the Hadley circulation. The

surface westerlies and surface trade winds are also marked, as are the highs and

lows of middle latitudes. Only the northern hemisphere is shown. The vertical

scale is greatly exaggerated.

is a straightforward consequence of the geometry of the earth: the annuallyaveraged incoming solar radiation per unit area of the earths surface is much

greater at the equator than at the poles, a difference that is compounded bythe fact that the polar regions are covered in ice and snow and thereforereflect much of the incoming radiation back to space. A little less obvious isthe fact that the tropical regions actually receive more energy from the sunthan they emit back to space, while the converse is true in high latitudes.Since both regions are, on an annual average, in equilibrium, there must bea process acting to transport excess energy from the tropics to make up thedeficit in high latitudes, as depicted schematically in Fig.8.1(left).

The implied transport of some 6 1015W see Fig.5.6 must beeffected by the atmospheric circulation, carrying warm air poleward and cold

air equatorward. (In fact, the ocean circulation also contributes, as discussedin Chapter 11.) As a result, the tropics are cooler, and polar regions warmer,than they would be in the absence of such transport. Thus, in this as in otherrespects, the atmospheric general circulation plays a key role in climate.

What are the motions that deliver the required transport? The gross


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features of the observed atmospheric circulation discussed in Chapter 5 are

depicted in Fig.8.2. The zonalfl

ow is strongly westerly aloft in middle lati-tudes, a fact that we can now, following Section 7.3, understand as a straight-forward consequence of the decrease of temperature with latitude. Surfacewinds are constrained to be weak by the action of friction near the ground,and thermal wind balance, Eq.(7.24), implies that a poleward decrease oftemperature is necessarily accompanied by increasing westerly winds withheight. Taken together, these two facts require a zonal flow in middle lat-itudes that increases from near zero at the ground to strong westerlies ataltitude, as schematized in Fig.7.19.

Although the near-surface winds are weak a few m s1 they nev-ertheless exhibit a distinct spatial distribution, with the zonal componentbeing easterly in the tropics and westerly in middle latitudes (Fig.7.28). Inthe presence of surface friction, the atmosphere must therefore be losingangular momentum to the ground in middle latitudes and gaining angularmomentum in the tropics. As Fig.8.1(right) illustrates, the angular momen-tum balance of the atmosphere thus requires a transport of westerly angularmomentum from low to middle latitudes in order to maintain equilibrium.

Even though the west-to-east circulation in the upper troposphere is thedominant component of the large-scale atmospheric flow, is cannot be respon-sible for the required poleward transports of heat and angular momentum,for which north-south flow is needed. As we saw in Chapter 5 (Fig.5.21) thereis indeed a mean circulation in the meridional plane, which is dominated bythe Hadley circulation of the tropical atmosphere with, on an annual aver-age, mean upwelling near the equator, poleward flow aloft, subsidence in thesubtropics and equatorward return flow near the surface. This circulationtransports heat and angular momentum poleward, as required, within thetropics; however, the meridional circulation becomes much weaker in middle

latitudes and so cannot produce much transport there. We saw in Section5.4.1 that most of the north-south flow in the extratropical atmosphere takesthe form of eddies, rather than a mean overturning as in the tropics. As wewill shall see, it is these eddies that produce the poleward transports in theextra-tropics.


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8.2. A MECHANISTIC VIEW OF THE CIRCULATION 253

8.2 A mechanistic view of the circulation

To what extent can we explain the main features of the observed generalcirculation on the basis of the fluid dynamics of a simple representation ofthe atmosphere driven by latitudinal gradients in solar forcing? The emphasishere is on simple. In reality, the Earths surface is very inhomogeneous:there are large mountain ranges that disturb the flow and large contrasts (e.g.,in temperature and in surface roughness) between oceans and continents.In the interests of simplicity, however, we shall neglect such variations insurface conditions. We also neglect seasonal and diurnal1 variations, andassume maximum solar input at the equator, even though the subsolar point(Fig.5.3) migrates between the two Tropics through the course of a year.

Thus, we shall consider the response of an atmosphere on a longitudinallyuniform, rotating, planet (that is otherwise like the Earth) to a latitudinalgradient of heating. We shall see that the gross features of the observedcirculation can indeed be understood on this basis; later, in Section 8.5, theshortcomings introduced by this approach will be briefly discussed.

Thus, we ask how an axisymmetric atmosphere responds to an axisym-metric forcing, with stronger solar heating at the equator than at the poles.It seems reasonable to assume that the response (i.e., the induced circula-tion) will be similarly axisymmetric. Indeed, this assumption turns out togive us a qualitatively reasonable description of the circulation in the tropics,but it will become evident that it fails in the extratropical atmosphere, where

the symmetry is broken by hydrodynamical instability.

8.2.1 The tropical Hadley circulation

If the Earth were not rotating, the circulation driven by the pole-equatortemperature difference would be straightforward, with warm air rising inlow latitudes and cold air sinking at high latitudes as first suggested byHadley and sketched in Fig.5.19. But, as seen in Figs.5.20 and 5.21, this isnot quite what happens. We do indeed see a meridional circulation in thetropics, but the sinking motion is located in the subtropics around latitudes of

1 In reality, of course, there are strong instantaneous longitudinal variations in solarforcing between those regions around local noon and those in nighttime. However, exceptvery near the surface, the thermal inertia of the atmosphere damps out these fluctuationsand temperature varies little between day and night. Hence neglect of diurnal variationsis in fact a reasonable approximation.


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Figure 8.3: Schematic of a ring of air blowing west east at speed u andlatitude . The ring is assumed to be advected by the poleward flow of the Hadleycirculation conserving angular momentum.

30. In fact, the following considerations tell us that one giant axisymmetricmeridional cell extending from equator to pole is not possible on a rapidlyrotating planet such as the Earth.

Consider a ring of air encircling the globe as shown in Fig.8.3, lyingwithin, and being advected by, the upper level poleward flow of the Hadleycirculation. Because this flow is, by assumption, axisymmetric and also be-cause friction is negligible in this upper level flow, well above the near-surfaceboundary layer, absolute angular momentum will be conserved by the ringas it moves around. The absolute angular momentum per unit mass is (re-call our discussion of angular momentum in the radial inflow experiment inSection 6.6.1)

A = r2 + ur ,

the first term being the contribution from planetary rotation, and the secondfrom the eastward wind u relative to the Earth, where r is the distance fromthe Earths rotation axis. Since r = a cos ,

A = a2 cos2 + ua cos . (8.1)

Now, suppose that u = 0 at the equator (Fig.5.20 shows that this is a rea-sonable assumption in the equatorial upper troposphere). Then the absoluteangular momentum at the equator is simply A0 = a

2. As the ring of air


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moves poleward, it retains this value and so when it arrives at latitude , its

absolute angular momentum isA = a2 cos2 + ua cos = A0 = a

2 .

Therefore the ring will acquire an eastward velocity

u () = (a2 a2 cos2 )

a cos = a

sin2

cos . (8.2)

Note that this is directly analogous to Eq.(6.23) of our radial inflow exper-iment, when we realize that r = a cos . Eq.(8.2) implies unrealisticallylarge winds far from the equator: at latitudes of (10, 20, 30), u() =

(14, 58, 130) ms1

, and of course the wind becomes infi

nite as

90

.On the grounds of physical plausibility, it is clear that such an axisymmetriccirculation cannot extend all the way to the pole in the way envisaged byHadley (and sketched in Fig.5.19): it must terminate somewhere before itreaches the pole. Just how far the circulation extends depends (according totheory) on many factors2.

Consider the upper branch of the circulation, as depicted in Fig.8.4. Nearthe equator, where f is small and the Coriolis effect is weak, angular mo-mentum constraints are not so severe and the equatorial atmosphere actsas if the Earth were rotating slowly. As air moves away from the equa-tor, however, the Coriolis parameter becomes increasingly large and, in the

northern hemisphere, turns the wind to the right, resulting in a westerlycomponent to the flow. At the poleward extent of the Hadley cell, then,we expect to find a strong westerly flow, as indeed we do see Fig.5.20.This subtropical jet is driven in large part by the advection from the equa-tor of large values of absolute angular momentum by the Hadley circulation,as is evident from Eq.(8.2) and depicted in Fig.8.4. Flow subsides on thesubtropical edge of the Hadley cell, sinking into the subtropical highs (veryevident in Fig.7.27), before returning to the equator at low levels. At theselow levels, the Coriolis acceleration, again turning the equatorward flow toits right in the northern hemisphere, produces the trade winds, northeast-

erly in the northern hemisphere (southeasterly in the southern hemisphere) see Fig.7.28. These winds are not nearly as strong as in the upper tro-posphere because they are moderated by friction acting on the near-surface

2 For example, if the Earth were rotating less (or more) rapidly, and other things beingequal, the Hadley circulation would extend further (or less far) poleward.


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Figure 8.4: Schematic of the Hadley circulation (showing only the Northern Hemi-spheric part of the circulation; there is a mirror image circulation south of the

equator). Upper level poleward flow induces westerlies; low level equatorward flow

induces easterlies

Figure 8.5: A schematic diagram of the Hadley circulation and its associatedzonal flows and surface circulation.


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flow. In fact, as discussed above, there must be low-level westerlies some-

where: in equilibrium, the net frictional drag (strictly, torque) on the entireatmosphere must be zero, or the total angular momentum of the atmospherecould not be steady. (There would be a compensating change in the angularmomentum of the solid Earth, and the length of the day would drift 3). So,as shown in Fig.8.5, the surface winds in our axisymmetric model would bewesterly at the poleward edge of the circulation cell, and eastward near theequator. This is similar to the observed pattern (see Fig.7.28, middle panel),but not quite the same: in reality the surface westerlies maximize near 50N,S, significantly poleward of the subtropical jet, a point to which we return inSection 8.4.2.

As sketched in Fig.8.5, the subtropical region of subsidence is warm (be-

cause of adiabatic compression) and dry (since the air aloft is much drierthan surface air; the boundary formed between this subsiding air and thecooler, moister near-surface air is the trade inversion noted in Chapter 4and within which the trade winds are located. Aloft, horizontal tempera-ture gradients within the Hadley circulation are very weak see Figs.5.7and 5.8 a consequence of very efficient meridional heat transport by thecirculation.

Experiment on the Hadley circulation: GFD Lab VIII revisited

A number of aspects of the Hadley circulation are revealed in Expt VIII whose experimental arrangement has already been described in Fig.7.12 inthe context of the thermal wind equation. The apparatus is just a cylindricaltank containing plain water, at the center of which is a metal can filled withice. The consequent temperature gradient (decreasing poleward) drivesmotions in the tank, the nature of which depends on the rotation rate of theapparatus. When slowly rotating, as in this experiment ( . 1 rpm yes,only 1 rotation of the table per minute: very slow!), we see the developmentof the thermal wind in the form of a strong eastward (i.e., super-rotating)flow in the upper part of the fluid which can be revealed by paper dotsfloating on the surface and dye injected in to the fluid as seen in Fig.7.13.

3 In fact the length of the day does vary, ever so slightly, due to angular momen-tum transfer between the atmosphere and underlying surface. For example on seasonaltimescales there are changes in the length of the day of about a millisecond. The lengthof the day changes from day to day by about 0.1ms! Moreover these changes can beattributed to exchanges of angular momentum between the earth and the atmosphere.


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Figure 8.6: The Hadley regime studied in GFD Lab VIII, Section 7.3.1. Bot-tom flow is revealed by the two outward spiralling streaks showing anti-cyclonic

(clockwise) flow sketched schematically in Fig.7.12(right); the black paper dots and

collar of dye mark the upper level flow and circulate cyclonically (anticlockwise).


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The azimuthal current observed in the experiment, which is formed in

a manner analogous to that of the subtropical jet by the Hadley circula-tion discussed above, is maintained by angular momentum advection by themeridional circulation sketched in Fig.7.12(right). Water rises in the outerregions, moves inward in the upper layers, conserving angular momentumas it does so, thus generating strong westerly flow, and rubs against thecold inner wall, becoming cold and descending. Potassium permanganatecrystals, dropped into the fluid (not too many!), settle on the bottom andgive an indication of the flow in the bottom boundary layer. In Fig.8.6 wesee flow moving radially outwards at the bottom and being deflected to theright, creating easterly flow in a direction opposite to that of the rotatingtable: note the two dark streamers moving outward and clockwise (opposite

to the sense of rotation and the upper level flow). This bottom flow is di-rectly analogous to the easterly and equatorward trade winds of the loweratmosphere sketched in Fig.8.4 and plotted from observations in Fig.7.28.

Quantitative study of the experiment (by tracking, for example, floatingpaper dots) shows that the upper level azimuthal flow does indeed conserveangular momentum, satisfying Eq.(6.23) quite accurately, just as in the radialinflow experiment. With = 0.1 s1 (corresponding to a rotation rateof 0.95rpm see Table 13.1, Appendix 13.4.1) Eq.(6.23) implies a hefty10cms1 if angular momentum were conserved by a particle moving from theouter radius r1 = 30 cm to an inner radius of 30cm. Note, however, that if were set 10 times higher, to 10rpm, then angular momentum conservationwould imply a speed of 1 m s1, a very swift current. We will see in thenext section that such swift currents are not observed if we turn up therotation rate of the table. Instead the azimuthal current breaks down intoeddying motions, inducing azimuthal pressure gradients and breaking angularmomentum conservation.

8.2.2 The extra-tropical circulation and GFD Lab XI:

baroclinic instability

Although the simple Hadley cell model depicted in Fig.8.5 describes the trop-ical regions quite well, it predicts that little happens in middle and highlatitudes. There, where the Coriolis parameter is much larger, the powerfulconstraints of rotation are dominant and meridional flow is impeded. Weexpect, however, (and have seen in Fig.5.8 and 7.20), that there are strong


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gradients of temperature in middle latitudes, particularly in the vicinity of

the subtropical jet. So, although there is little meridional circulation, there isa zonal flow in thermal wind balance with the temperature gradient: since Tdecreases poleward, Eq.(7.24) implies westerly winds increasing with height.This state of a zonal flow with no meridional motion is a perfectly valid equi-librium state: even though the existence of a horizontal temperature gradientimplies horizontal pressure gradients (tilts of pressure surfaces), the associ-ated force is entirely balanced by the Coriolis force acting on the thermalwind, as we discussed in Chapter 7.

Our deduction that the mean meridional circulation is weak outside thetropics is qualitatively in accord with observations (see Fig.5.21) but leavesus with two problems:

1. Poleward heat transport is required to balance the energy budget. Inthe tropics, the overturning Hadley circulation transports heat pole-ward, but no further than the subtropics. How is heat transportedfurther poleward if there is no meridional circulation?

2. Everyday observation tells us that a picture of the midlatitude at-mosphere as one with purely zonal winds is very wrong. If it weretrue, weather would be very predictable (and very dull).

Our axisymmetric model is, therefore, only partly correct. The predic-

tion of purely zonal flow outside the tropics is quite wrong. As we have seen,the midlatitude atmosphere is full of eddies, which manifest themselves astraveling weather systems.4 Where do they come from? In fact, as we shallnow discuss and demonstrate through a laboratory experiment, the extra-tropical atmosphere is hydrodynamically unstable, the flow spontaneously

4 Albert Defant (1884 1974) German Professor of Meteorology andOceanography, who made important contributions to the theory of the general circulationof the atmosphere. He was the first to liken the meridional transfer of energy between thesubtropics and pole to a turbulent exchange by large-scale eddies.


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breaking down into eddies through a mechanism known as baroclinic insta-

bility

5

. These eddies readily generate meridional motion and, as we shall see,effect a meridional transport of heat.

Baroclinic instability

GFD Lab XI baroclinic instability in a dishpan To introduce ourdiscussion of the breakdown of the thermal wind through baroclinic instabil-ity we describe a laboratory experiment of the phenomenon. The apparatus,sketched in Fig.7.12, is identical to that of Lab VIII used to study the ther-mal wind and the Hadley circulation. In the former experiments the tablewas rotated very slowly, at a rate of . 1 rpm. This time, however, the

table is rotated much more rapidly, at

10 rpm, representing the consid-erably greater Coriolis parameter found in middle latitudes. At this higherrotation rate something remarkable happens. Rather than a steady axisym-metric flow, as in the Hadley regime shown in Fig.8.6, a strongly eddyingflow is set up. The thermal wind remains, but breaks down through insta-bility, as shown in Fig.8.7. We see the development of eddies which sweep(relatively) warm fluid from the periphery to the cold can in one sector of thetank (e.g. A in Fig.8.7) and, simultaneously, carry cold fluid from the can tothe periphery at another (e.g. B in Fig.8.7). In this way a radially-inwardheat transport is achieved, offsetting the cooling at the centre caused by themelting ice.

For the experiment shown we observe three complete wavelengths aroundthe tank in Fig.8.7. By repeating the experiment but at different values of, we observe that the scale of the eddies decreases and the flow becomesincreasingly irregular as is increased. These eddies are produced by thesame mechanism at work in the creation of atmospheric weather systemsshown in Fig.7.20 and discussed below.

Before going on we should emphasize that the flow in Fig.8.7 does notconserve angular momentum. Indeed, if it did, as estimated at the end ofSection 8.2.1, we would observe very strong horizontal currents near the icebucket, of order 1 m s1 rather than a few cm s1 as seen in the experiment.

This, of course, is exactly what Eq.(8.2) is saying for the atmosphere: if ringsof air conserved angular momentum as they move in axisymmetric motion

5 In a baroclinic fluid = (p, T) and so there can be gradients of density (and there-fore of temperature) along pressure surfaces. This should be contrasted to a barotropicfluid ( = (p)) in which no such gradients exist.


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Figure 8.7: (Top) Baroclinic eddies in the eddy regime viewed from the side.

(Bottom) View from above. Eddies drawfl

uid from the periphery in toward thecentre at point A and vice-versa at point B. The eddies are created by the insta-

bility of the thermal wind induced by the radial temperature gradient due to the

presence of the ice bucket at the centre of the tank. The diameter of the ice bucket

is 15cm.


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Figure 8.8: (top) In middle latitudes eddies transport warm air poleward and up-ward and cold air equatorward and downward. Thus, the eddies tend to stir the

atmosphere laterally reducing the equator-to-pole temperature contrast. (middle)

To the west of the L cold air is carried in to the tropics. To the east, warm air is

carried toward the pole. The resulting cold fronts (marked by triangles) and warm

fronts (marked by semi-circles) are indicated. (bottom) Sections through the cold

front, a a0, and the warm front, b b0, respectively.


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from equator to pole, then we obtain unrealistically large winds. Angular

momentum is not conserved because of the presence of zonal (or, in our tankexperiment, azimuthal) pressure gradient associated with eddying motion(see Question 6 of Chapter 6).

Middle latitude weather systems The process of baroclinic instabilitystudied in the above laboratory experiment is responsible for the ubiquitouswaviness of the midlatitude flow in the atmosphere. As can be seen in theobservations shown on Figs.5.22, 7.4 and 7.20, these waves often form closededdies, especially near the surface, where they are familiar as the high and lowpressure systems associated with day-to-day weather. In the process, theyalso affect the poleward heat transport required to balance global energy

budget (see Fig.8.1). The manner in which this is achieved on the planetaryscale is sketched in Fig.8.8: eddies stir the atmosphere carrying cold airequatorwards and warm air polewards, thereby reducing the equator-to-poletemperature contrast. To the west of the low (marked L) in Fig.8.8, cold airis carried into the tropics. To the east, warm air is carried toward the pole,but since poleward flowing air is ascending (remember the large-scale slope ofthe surfaces shown in Fig.5.8) it tends to leave the surface. Thus we get aconcentrated gradient of temperature near point 1 where the cold air pushesinto the warm air and a second, less marked, concentration where the warmair butts into cold at point 2. We can thus identify cold and warm fronts,

respectively, as marked in the centre panel of Fig.8.8. Note that a triangleis used to represent the sharp cold front and a semicircle to represent thegentler warm front. In the bottom panel we present sections through coldfronts and warm fronts respectively.

Timescales and length scales We have demonstrated by laboratory ex-periment that a current in thermal wind balance can, and almost alwaysdoes, become hydrodynamically unstable, spawning meanders and eddies.More detailed theoretical analysis6 shows that the lateral scale of the eddies

6Detailed analysis of the space-scales and growth rates of the instabilities that sponta-neously arise on an initially zonal jet in thermal wind balance with a meridional temper-ature gradient show that:

1. the Eady growth rate of the disturbance et is given by = 0.31 UL

= 0.31 fN

dudz

using Eq.(7.23) and U = dudzH (see Gill, 1982). Inserting typical numbers for the

troposphere N = 102 s1, dudz

= 2 103 s1, f = 104 s1 we find that


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that form, Leddy (as measured by, for example, the typical lateral scale of a

low pressure systems in the surface analysis shown in Fig.7.25, or the swirlsof dye seen in our tank experiment, Fig.8.7) is proportional to the Rossbyradius of deformation discussed in Section 7.3.4:

Leddy L (8.3)

where L is given by Eq.(7.23).

The time-scale of the disturbance is given by:

Teddy L

U(8.4)

where U = dudz

H is the strength of the upper level flow, dudz

is the thermalwind given by Eq.(7.17) and H is the vertical scale of the flow. Eq.(8.4) isreadily interpretable as the time it takes a flow moving at speed U to travela distance L and is proportional to the (inverse of) the Eady growth rateafter Eric Eady, a pioneer of the theory of baroclinic instability.

In our baroclinic instability experiment, we estimate a deformation radiusof some 10cm roughly in accord with the observed scale of the swirls in Fig.8.7(to determine a scale, note that the diameter of the ice can in the figure issome 15cm). Typical flow speeds observed by the tracking of paper dots

floating at the surface of the fluid are about 1c m s

1. Thus Eq.(8.4) suggestsa time-scale ofTeddy 7.0 s or, roughly, one rotation period. Consistent withEq.(8.3), the eddy scale decreases with increased rotation rate.

Applying the above formulae to the middle troposphere where f 104 s1,N 102 s1 (see Section 4.4), H 7 km (see Section 3.3) and U 1 0 m s1,we find that L =

NHf 700km and Teddy

700 km10ms1

1 d. These estimatesare roughly in accord with the scales and growth rates of observed weathersystems.

' 105 s1, an efolding time-scale of 1 day.

2. the wavelength of the fastest growing disturbance is 4L where L is given byEq.(7.23). This yields a wavelength of 2800 km ifL = 700 km. The circumferenceof the earth at 45N is 21000 km and so about 7 synoptic waves can fit around theearth at any one time. This is roughly in accord with observations; see, for example,Fig.5.22.


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We have not yet discussed the underlying baroclinic instability mechanism

and its energy source. In order to do so we now consider the energetics of,and the release of potential energy from, a fluid.

8.3 Energetics of the thermal wind equation

The immediate source of kinetic energy for the eddying circulation observedin our baroclinic instability experiment and in the middle-latitude atmosphere,is the potential energy of the fluid. In the spirit of the energetic discussion ofconvection developed in Section 4.2.3, we now compute the potential energyavailable for conversion to motion. However, rather than, as there, consider-ing the energy of isolated fluid parcels, here we focus on the potential energyof the whole fluid. It will become apparent that not all the potential energyof a fluid is available for conversion to kinetic energy. We need to iden-tify that component of the potential energyknown as available potentialenergythat can be released by a redistributions of mass of the system.

To keep things as simple as possible, we will first focus on an incompress-ible fluid, such as the water in our tank experiment. We will then go on toaddress the compressible atmosphere.

8.3.1 Potential energy for a fluid system

We assign to a fluid parcel of volume dV = dx dy dz and density , apotential energy of gz (parcel mass) = gz dV. Then the potential energyof the entire fluid is

P E = g

Zz dV , (8.5)

where the integral is over the whole system. Clearly, P E is a measure of thevertical distribution of mass.

Energy can be released and converted to kinetic energy only if some re-arrangement of the fluid results in a lower total potential energy. Such re-arrangement is subject to certain constraints, the most obvious of which isthat the total mass

M =Z

dV , (8.6)

cannot change. In fact, we may use Eq.(8.6) to rewrite Eq.(8.5) as

P E = gMhzi (8.7)


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8.3. ENERGETICS OF THE THERMAL WIND EQUATION 267

Figure 8.9: Reduction in the available potential energy of a two-layer fluid movingfrom an initial sate in which the interface is tilted (left) to the final state in which

the interface is horizontal (right). Dense fluid is shaded. The net effect of the

rearrangement is to exchange heavy fluid downward and light fluid upward such

that, in the wedge B, heavy fluid is replaced by light fluid, while the oppositeoccurs in the wedge A.

where

hzi =

Rz dVR dV

=

Rz dV

M(8.8)

is the mass-weighted mean height of the fluid. We can think of hzi as theheight of the center of mass. It is evident from Eq.(8.7) that potentialenergy can be released only by lowering the center of mass of the fluid.

8.3.2 Available potential energy

Consider again the stratified incompressible fluid discussed in Section 4.2.2.Suppose, first, that /z < 0 (stably stratified) and that there are no hor-izontal gradients of density, so is a function of height only, as depicted inFig.4.5. In Section 4.2.3 we considered the energetic implications of adia-batically switching the position of two parcels, initially at heights z1 and z2,

with densities 1 and 2, respectively, with z2 > z1, and 2 < 1. Since isconserved under the displacement (remember we are considering an incom-pressible fluid here), the final state has a parcel with density 1 at z2, andone with 2 at z1. Thus, density has increased at z2 and decreased at z1. Thecenter of mass has therefore been raised and so P E has been increased by


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the rearrangement: none can be released into kinetic energy. This, of course,

is one way of understanding why the stratifi

cation is stable. Even thoughsuch a fluid has nonzero potential energy (as defined by (8.7)), this energyis unavailable for conversion to kinetic energy: it cannot be reduced by anyadiabatic rearrangement of fluid parcels. Therefore, the fluid has availablepotential energy only if it has nonzero horizontal gradients of density. 7

Consider now the density distribution sketched in Fig.8.9(left); a two-layer fluid is shown with light fluid over heavy with the interface betweenthe two sloping at angle . This can be considered to be a highly idealizedrepresentation of the radial density distribution in our tank experiment GFDLab XI with heavier fluid (shaded) to the right (adjacent to the ice bucket)and light fluid to the left. Let us rearrange the fluid such that the interface

becomes horizontal as shown. Now all fluid below the dashed horizontal lineis dense, and all fluid above is light. The net effect of the rearrangement hasbeen to exchange heavy fluid downward and light fluid upward such that, inthe wedge B, heavy fluid has been replaced by light fluid, while the oppositehas happened in the wedge A. (Such a rearrangement can be achieved in morethan one way, as we shall see.) The center of mass has thus been lowered,and the final potential energy is therefore less than that in the initial state:potential energy has been released. Since no further energy can be releasedonce the interface is horizontal, the difference in the potential energy betweenthese two states defines the available potential energy (AP E) of the initialstate.

The important conclusions here are that available potential energy isassociated with horizontal gradients of density and that release of that energy(and, by implication, conversion to kinetic energy of the motion) is effectedby a reduction in those gradients.

7 Eric Eady (1918-1967). A brilliant theorist who was a forecasterin the early part of his career, Eady expounded the theory of baroclinic instability in awonderfully lucid paper Eady (1949) which attempted to explain the scale, growthrate and structure of atmospheric weather systems.


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The above discussion can be made quantitative as follows. Consider again

Fig.8.9 in which the height of the interface between the twofl

uids is givenby h(y) = 12

H + y and, to keep things simple, we restrict the slope of theinterface < H

2Lto ensure that it does not intersect the lower and upper

boundaries at z = 0, z = H respectively. Direct integration of Eq.(8.5) forthe density distribution shown in Fig.8.9, shows that the potential energyper unit length in the xdirection is given by (as derived in the Appendix13.1.2)

P =

ZH0

ZLL

gz dy dz = H2L1

g +

g0

4

+

1

31g

02L3

where g0

= g

(12)

1 is the reduced gravity of the system. P is at its absoluteminimum, Pmin, when = 0, and given by the first term on the lhs of theabove. When = 0, no rearrangement offluid can lower the center of mass,since all dense fluid lies beneath all light fluid. When 6= 0, however, Pexceeds Pmin, and the fluid has available potential energy in the amount

AP E = P Pmin =1

31g

02L3 . (8.9)

Note that AP E is independent of depth H because all the AP E is associatedwith the interfacial slope; only fluid motion in the vicinity of the interfacechanges potential energy, as is obvious by inspection of Fig.8.9.

It is interesting to compare the AP E associated with the tilted interfacein Fig.8.9 with the kinetic energy associated with the thermal wind currentwhich is in balance with it. Making use of Margules formula, Eq.(7.21), thegeostrophic flow of the upper layer is given by u2 = g

0 2

if the lower layer isat rest. Thus the kinetic energy of the flow per unit length is, on evaluation:

KE =

ZHh

ZLL

1

22u

22 dy dz =

1

22

g022

(2)2HL (8.10)

The ratio of AP E to KE is thus:

AP EKE

= 23

L

L

2

(8.11)

where we have set 12 1 and used Eq.(7.23) to express the result in terms

of the deformation radius L.


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Figure 8.10: Release of available potential energy in a two-component fluid. (left)Non-rotating (or very slowly rotating) case: azimuthally uniform overturning.

(right) Rapidly rotating case: sloping exchange in the wedge of instability by

baroclinic eddies.

Eq.(8.11) is a rather general result for balanced flows the ratio ofAP Eto KE is proportional to the square of the ratio of L, the lateral scale overwhich the flow changes, to the deformation radius L. Typically, on the large-scale, L is considerably larger than L and so the potential energy stored inthe sloping density surfaces greatly exceeds the kinetic energy associatedwith the thermal wind current in balance with it. For example, T varies onhorizontal scales of L 5000 km in Fig.5.7. In Section 8.2.2 we estimated

that L 700km: thus (L/L)2 ' 50. This has important implications foratmospheric and (as we shall see in Chapter 10) oceanic circulation8.

8.3.3 Release of Available Potential Energy in baro-

clinic instability

In a non-rotating fluid, the release of AP E is straightforward and intuitive.If, at some instant in time, the interface of our two-component fluid weretilted as shown in Fig.8.10(left), we would expect the interface to collapsein the manner depicted by the heavy arrows in the figure. In reality, the

interface would overshoot the horizontal position, and the fluid would slosh

8In fact in the ocean

LL

2' 400 telling us that the available potential energy stored

in the main thermocline exceeds the kinetic energy associated with ocean currents by afactor of 400. (See Section 10.5.)


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around until friction brought about a motionless steady state with the in-

terface horizontal. The AP E released by this process wouldfi

rst have beenconverted to kinetic energy of the motion, and ultimately lost to frictionaldissipation. In this case, there would be no circulation in the long term, butif external factors such as heating at the left of Fig.8.10(left) and cooling atthe right were to sustain the tilt in the interface, there would be a steadycirculation (shown by the light arrows in the figure) in which the interface-flattening effects of the circulation balance the heating and cooling.

In a rotating fluid, things are not so obvious. There is no necessity fora circulation of the type shown in Fig.8.10(left) to develop since, as we sawin Section 7.3.5, an equilibrium state can be achieved in which forces associ-ated with the horizontal density gradient are in thermal wind balance witha vertical shear of the horizontal flow. In fact, rotation actively suppressessuch circulations. We have seen that if is sufficiently small, as in theHadley circulation experiment GFD Lab VIII (section 8.2.1), such a circula-tion is indeed established. At high rotation rates, however, a Hadley regimeis not observed; nevertheless, AP E can still be released, albeit by a verydifferent mechanism. As we discussed in Section 8.2.2, the zonal state wehave described is unstable to baroclinic instability. Through this instability,azimuthally asymmetric motions are generated, within which fluid parcelsare exchanged along sloping surfaces as they oscillate from side to side, assketched in Fig.8.8. The azimuthal current is therefore not purely zonal,but wavy, as observed in the laboratory experiment shown in Fig.8.7. Theonly way light fluid can be moved upward in exchange for downward motionof heavy fluid is for the exchange to take place at an angle that is steeperthan the horizontal, but less steep than the density surfaces, i.e., within the

so-called wedge of instability, as shown in Fig.8.10(right). Thus, AP E isreduced, not by overturning in the radial plane, but by fluid parcels oscil-lating back and forth within this wedge: light (warm) fluid moves upwardsand radially inwards at one azimuth while, simultaneously, heavy (cold) fluidmoves downwards and outwards at another as sketched in Fig.8.8. This is


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the mechanism at work in GFD Lab XI.9

8.3.4 Energetics in a compressible atmosphere

So far we have discussed energetics in the context of an incompressible fluidsuch as water. However, it is not immediately clear how we can apply con-cepts such as available potential energy to a compressible fluid such as theatmosphere. Parcels of incompressible fluid can do no work on their sur-roundings and T is conserved in adiabatic rearrangement of the fluid: po-tential energy is all one needs to consider. However, air parcels expand andcontract during the redistribution of mass, doing work on their surroundingsand changing their T and hence internal energy. Thus we must also considerchanges in internal energy. It turns out that, for an atmosphere in hydro-static balance and with a flat lower boundary, it is very easy to extend thedefinition Eq.(8.5) to include the internal energy. The internal energy (IE)of the entire atmosphere (assuming it to be a perfect gas) is, if we neglectsurface topography (see Appendix 13.1.3)

IE = cv

ZT dV =

cvR

g

Zz dV . (8.12)

Note the similarity with Eq.(8.5): in fact, for a diatomic gas like the at-

mosphere, cv/R = (1

) / = 5/2, so its internal energy (neglecting topo-graphic effects) is 2.5 times greater than its potential energy. The sum ofinternal and potential energy is conventionally (if somewhat loosely) referredto as total potential energy, and given by

9 Jule Gregory Charney (1917-1981), American meteorologist and MITProfessor. He made wide-ranging contributions to the modern mathematical theory ofatmospheric and oceanic dynamics, including the theory of large-scale waves, and of baro-clinic instability. He was also one of the pioneers of numerical weather prediction, produc-ing the first computer forecast in 1950.


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Figure 8.11: Air parcels 1 and 2 are exchanged along paths marked A and Bconserving potential temperature . The continuous lines are observed surfaces see Fig. 5.8. The tilted dotted line is parallel to the local .

T P E = P E+ IE =g

Zz dV =

g

Mhzi (8.13)

where, as before, hzi is defined by Eq.(8.8) and M by Eq.(8.6).We can apply the ideas discussed in the previous section to a compressible

fluids if, as discussed in Chapter 4, we think in terms of the distribution ofpotential temperature, rather than density, since the former is conservedunder adiabatic displacement.

Consider Fig.8.11 in which the distribution of in the atmosphere, increasing upward and equatorward as in Fig.5.8, is schematically shown.Again, we suppose that two air parcels are exchanged adiabatically. Supposefirst that we exchange parcel 1, with = 1, with a second parcel with = 2 > 1, vertically along path A. It is clear that the final potential energywill be greater than the initial value because the parcel moving upward iscolder than the parcel it replaces. This is just the problem considered in our

study of stability conditions for dry convection in Section 4.3. If parcels areexchanged along a horizontal path or along a path in a surface of constant (dotted lines in the figure), there will be no change in potential energy.But consider exchanging parcels along path B in Fig.8.11, which has a slopebetween that of the surface and the horizontal. The parcel moving upwards


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is now warmer than the parcel moving downwards and so potential energy

has been reduced by the parcel rearrangement. In other words the centerof gravity of the fluid is lowered. In the process, a lateral transport of heatis achieved from cold to warm. It is this release of APE that powers theeddying motion. As discussed in Section 8.3.3, fluid parcels are exchangedin this wedge of instability as they oscillate from side to side, as sketchedin Fig.8.8.

8.4 Large-scale atmospheric heat and momen-

tum budget

8.4.1 Heat transport

We have seen two basic facts about the atmospheric energy budget:

1. There must be a conversion of energy into kinetic energy, from theavailable potential energy gained from solar heating, which requiresthat motions develop which transport heat upward (thereby tending tolower the atmospheric center of mass).

2. Accompanying this upward transport, there must be a poleward trans-port of heat from low latitudes; this transport cools the tropics andheats the polar regions, thus balancing the observed radiative budget.

It is fairly straightforward to estimate the rate of poleward heat transportby atmospheric motions. Consider the northward flow at latitude acrossan elemental area dA, of height dz and longitudinal width d, so dA =a cos d dz. The rate at which mass is flowing northward across dA isv dA, and the associated flux of energy is vE dA, where

E = cp

T + gz + Lq+1

2u u (8.14)

is the moist static energy, Eq.(4.27) (made up of the dry static energycpT + gz, and the latent heat content Lq) and the kinetic energy density12u u. If we now integrate across the entire atmospheric surface at that


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8.4. LARGE-SCALE ATMOSPHERIC HEAT AND MOMENTUM BUDGET275

Figure 8.12: Schematic of the transport of (left) energy and (right) momentumby the atmospheric general circulation. Transport occurs through the agency of

the Hadley circulation in the tropics, and baroclinic eddies in middle latitudes

see also Fig.8.1.

latitude, the net energy flux is

H

atmos =

ZZvE dA

= a cos Z2

0Z

0

vE dz d

=a

gcos

Z20

Zps0

vE dp d , (8.15)

(using hydrostatic balance to replace dz by dp/) where ps is surfacepressure. It turns out that the kinetic energy component of the net transportis negligible (typically less than 1% of the total), so we only need to considerthe first three terms in Eq.(8.14).

Let us first consider energy transport in the tropics by the Hadley cir-culation which (in the annual mean) has equatorward flow in the lower tro-posphere and poleward flow in the upper troposphere. We may write

H

tropics =2a

gcos

Zps0

v (cpT + gz + Lq) dp . (8.16)

where we have assumed that the tropical atmosphere is independent of lon-gitude .


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Since temperature decreases with altitude, the heat carried poleward by

the upper troposphericfl

ow is less than that carried equatorward at loweraltitudes. Therefore the net heat transport the first term in Eq.(8.16) isequatorward: the Hadley circulation carries heat toward the hot equator fromthe cooler subtropics! This does not seem very sensible. However, now let usadd in the second term in Eq.(8.16), the potential energy density gz, to obtainthe flux of dry static energy v (cpT + gz). Note that the vertical gradient ofdry static energy is (cpT + gz) /z = cpT/z + g. For an atmosphere(like ours) that is generally stable to dry convection, T/z + g/cp > 0 (asdiscussed in Section 4.3.1) and so the dry static energy increases with height.Therefore the poleward flow at high altitude carries more dry static energypoleward than the low-level flow takes equatorward, and the net transport is

poleward, as we might expect.

To compute the total energy flux we must include all the terms in Eq.(8.16).The latent heat flux contribution is equatorward, since (see Fig.5.15) q de-creases rapidly with height, and so the equatorward flowing air in the lowertroposphere carries more moisture than the poleward moving branch aloft.In fact, this term almost cancels the dry static energy flux. This is becausethe gross vertical gradient of moist static energy cpT+ gz + Lq a first inte-gral of Eq.(4.26) under hydrostatic motion is weak in the tropics. This is

just another way of saying that the tropical atmosphere is almost neutral tomoist convection, a state that moist convection itself helps to bring about.

This is clearly seen in the profiles of moist potential temperature shown inFig.5.9 of Chapter 5. Thus the equatorward and poleward branches of thecirculation carry almost the same amount in each direction. In the net, then,the annually averaged energy flux by the Hadley cell is poleward, but weaklyso. In fact, as can be seen in Fig.8.13, the heat transport by the ocean ex-ceeds that of the atmosphere in the tropics up to 15 or so, particularly inthe northern hemisphere.

The relative contributions of the various components of the energy fluxare different in the extra-tropics, where the mean circulation is weak and es-sentially all the transport is effected by midlatitude eddies. In these motions,

the poleward and equatorward flows occur at almost the same altitude, andso the strong vertical gradients of gz and Lq are not so important. Whilethese components of the energy flux are not entirely negligible (and must beaccounted for in any detailed calculation), they are relatively small and sowe can use the heat flux alone to obtain a reasonable estimate of the net flux.


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8.4. LARGE-SCALE ATMOSPHERIC HEAT AND MOMENTUM BUDGET277

Figure 8.13: The Ocean (thin) and Atmospheric (dotted) contributions to the to-tal northwards heat flux (thick) based on the NCEP reanalysis (in PW= 1015W)by (i) estimating the net surface heat flux over the ocean (ii) the associated oceanic

contribution, correcting for heat storage associated with global warming and con-

straining the ocean heat transport to be -0.1 PW at 68S (iii) deducing the at-

mospheric contribution as a residual. The total meridional heat flux, as in Fig.5.6,

is also plotted (thick). From Trenberth and Caron (2001).


30/38


So in middle latitudes we can represent Eq.(8.15) by

Hmidlat 2 acp

gcos ps [v][T]

where [v] is a typical northward wind velocity (dominated by the eddy compo-nent in middle latitudes) and [T] is the typical magnitude of the temperaturefluctuations in the presence of the eddiesthe temperature difference be-tween equatorward and poleward flowing air. Given a = 6371 km, cp = 1005J kg1 K1, g = 9.8 1 m s2, ps ' 10

5 Pa, then if we take typical values of

[v] ' 10m s1 and [T] ' 3 K at a latitude of 45, we estimate H

midlat 8PW. As discussed earlier in Section 5.1.3 and in more detail below, this is ofthe same order as implied by the radiative imbalance.

In cartoon form, our picture of the low- and high-latitude energy balanceis therefore as shown in Fig.8.12(left). In the tropics energy is transportedpoleward by the Hadley circulation; in higher latitudes eddies are the prin-cipal agency of heat transport.

The results of more complete calculations, making use of top of the at-mosphere radiation measurements and analyzed atmospheric fields, of heatfluxes in the atmosphere and ocean are shown in Fig.8.13. The bulk of therequired transport is carried by the atmosphere in middle and high latitudes,but the ocean makes up a considerable fraction, particularly in the tropicswhere (as we have seen) atmospheric energy transport is weak. The role ofthe ocean in meridional heat transport, and the partition of heat transportbetween the atmosphere and ocean, are discussed in some detail in Section11.5.2.

8.4.2 Momentum transport

In addition to transporting heat, the general circulation also transports an-gular momentum. In the Hadley circulation, as we have seen, the upper,poleward-flowing, branch is associated with strong westerly winds: thus,westerly momentum is carried poleward. The lower branch, on the otherhand, is associated with easterlies that are weakened by surface friction;

the equatorward flow carries weak easterly momentum equatorward. Thenet effect is a poleward transport of westerly angular momentum. Midlati-tude eddies also transport angular momentum (albeit for less obvious and, infact, quite subtle reasons, as sketched in Fig.8.14), again mostly transportingeastward angular momentum poleward. This has the effect of modifying the


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8.5. LATITUDINAL VARIATIONS OF CLIMATE 279

Figure 8.14: Circular eddies on the left are unable to affect a meridional transferof momentum. The eddies on the right, however, by virtue of their banana

shape, transfer westerly momentum northwards south of, and southwards north

of, the mid-line. Weather systems transport westerly momentum from the tropics

(uv > 0) towards higher latitudes, as required in Fig.8.12(right), by trailing theirtroughs down in to the tropics (see, e.g., the H2O distribution shown in Fig.3), as

in the southern half of the banana-shaped eddy sketched on the right.

surface winds from that sketched schematically in Fig.8.5, by shifting the lowlatitude westerlies into middle latitudes.

The atmospheric angular momentum budget may therefore be depictedas in Fig.8.12(right). Because there is a net export of momentum out of lowlatitudes, there must be a supply of momentum into this region; the onlyplace it can come from is the surface, through friction acting on the lowlevel winds. In order for the friction to supply angular momentum to theatmosphere (and yet act as a brake on the low level winds) the low levelwinds in the tropics must be easterly, consistent with that deduced for theHadley circulation here. The balancing loss of westerly momentum fromthe atmosphere which must be associated with drag on the near-surfacewesterlies is located in middle latitudes and balances the supply of westerlymomentum via eddy transport.

8.5 Latitudinal variations of climate

Putting everything together, we can depict the atmospheric wind systems inthe upper and lower troposphere schematically as in Fig.8.15.


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Figure 8.15: Schematic of the global distributions of atmospheric winds in theupper (top) and lower (bottom) troposphere together with the primary latitudinal

climate zones.


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8.5. LATITUDINAL VARIATIONS OF CLIMATE 281

As we have remarked at various points along the way in this chapter, the

structure of the circulation dictates more than just the pattern of winds. Inthe near-equatorial regions, the convergence of the trade winds is associatedwith frequent and intense rainfall, as is characteristic of the deep tropics; itis here, for example, that the worlds great tropical rainforests are located.Around the subtropics (20 30 latitude), in the warm, dry, descendingbranch of the Hadley circulation, the climate is hot and dry: this is the sub-tropical desert belt. Poleward of about 30, where baroclinic eddies dominatethe meteorology, local winds vary from day to day, though they are predomi-nantly westerly, and the weather is at the mercy of the passing eddies: usuallycalm and fine in anticyclonic high pressure systems, and frequently wet andstormy in cyclonic, low pressure systems.

At least from a big picture viewpoint, we have thus been able to deducethe observed pattern of atmospheric winds, and the division of the world intomajor climate zones, through straightforward consideration of the propertiesof a dynamic atmosphere on a longitudinally uniform rotating planet subjectto longitudinally independent differential heating. We cannot proceed muchfurther without taking account of seasonal variations, and of longitudinalasymmetry of the Earth, which we do not have the space to develop here.However, we will briefly note some of the major differences between oursimple picture and observed circulation.

During the course of a year, the pattern of solar forcing migrates, northin northern summer, south in southern summer, and this has a significantimpact on the atmospheric response. Our picture of the typical Hadleycirculation (Fig.8.5) inevitably shows symmetry about the equator. In reality,the whole circulation tends to shift seasonally such that the upwelling branchand associated rainfall are found on the summer side of the equator (seeFig.5.21). The degree to which this happens is strongly controlled by localgeography: seasonal variations over the oceans, whose temperature variesrelatively little through the year, are weak, while they are much strongerover land. The migration of the main area of rainfall is most dramatic inthe region of the Indian Ocean, where intense rain moves onto the Asiancontinent during the summer monsoon.

While prevailing westerlies and migrating storms characterize most of theextratropical region throughout the year, there are seasonal variations (gener-ally with more storms in winter, when the temperature gradients are greater,than in summer) and strong longitudinal variations, mostly associated withthe distribution of land and sea. Longitudinal variations in climate can be


34/38


dramatic: consider the contrast across the Atlantic Ocean, where the Janu-

ary mean temperature at Goose Bay, Newfoundland (at about 53

N) is about30C colder than that at Galway, at almost the same latitude on the westcoast of Ireland. We cannot explain this with our longitudinally symmet-ric model of the world, but the explanation is nevertheless straightforward.Prevailing winds bring cold continental air to the coast of Newfoundland,but mild oceanic air to the west coast of Ireland. The North Atlantic Oceanis much warmer than the continents at the same latitude in winter, partlybecause of the large heat capacity of the ocean, and partly because of heattransport by the ocean circulation. The ocean circulation and its role inclimate is the subject of the following Chapters.

8.6 Further reading

An interesting historical account of evolving perspectives on the general cir-culation of the atmosphere is given in Lorenz (1967). Chapter 6 of Hartmann(1994) presents a thorough discussion of the energy and angular budgets ofthe general circulation; a more advanced treatment can be found in Chapter10 of Holton (2004).

8.7 Problems

1. Consider a zonally symmetric circulation (i.e., one with no longitudi-nal variations) in the atmosphere. In the inviscid upper troposphere,one expects such a flow to conserve absolute angular momentum, i.e.,DA/Dt = 0, where A = a2 cos2 + ua cos is the absolute angularmomentum per unit mass see Eq.(8.1) where is the Earth ro-tation rate, u the eastward wind component, a the Earths radius, and latitude.

(a) Show, for inviscid zonally symmetric flow, that the relation DA/Dt =0 is consistent with the zonal component of the equation of mo-

tion (using our standard notation, with Fx the x-component ofthe friction force per unit mass)

Du

Dt f v =

1

p

x+ Fx ,


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8.7. PROBLEMS 283

Figure 8.16: A schematic diagram of the Hadley circulation rising at 10S, crossing

the equator and sinking at 20

N.

in (x,y,z) coordinates, where y = a (see Fig.6.19).

(b) Use angular momentum conservation to describe how the existenceof the Hadley circulation explains the existence of both the sub-tropical jet in the upper troposphere and the near-surface TradeWinds.

(c) If the Hadley circulation is symmetric about the equator, and itsedge is at 20 latitude, determine the strength of the subtropical

jet.

2. Consider the tropical Hadley circulation in northern winter, as shownin Fig.8.16. The circulation rises at 10S, moves northward across theequator in the upper troposphere, and sinks at 20N. Assuming thatthe circulation, outside the near-surface boundary layer, is zonally sym-metric (independent of x) and inviscid (and thus conserves absoluteangular momentum about the Earths rotation axis), and that it leavesthe boundary layer at 10S with zonal velocity u = 0, calculate thezonal wind in the upper troposphere at (a) the equator, (b) at 10N,and (c) at 20N.

3. Consider what would happen if a force toward the pole were appliedto the ring of air considered in Q.1 if it conserved its absolute angularmomentum, A. Calculate the implied relationship between a smalldisplacement and the change in the speed of the ring u. How many


36/38


kilometers northwards does the ring have to be displaced in order to

change its relative velocity 10m s

1

? How does your answer depend onthe equilibrium latitude? Comment on your result.

4. An open dish of water is rotating about a vertical axis at 1 revolutionper minute. Given that the water is 1 C warmer at the edges thanat the center at all depths, estimate, under stated assumptions andusing the data below, typical azimuthal flow speeds at the free surfacerelative to the dish. Comment on, and give a physical explanation for,the sign of the flow.

How much does the free surface deviate from its solid body rotationform?

Briefly discuss ways in which this rotating dish experiment is a usefulanalogue of the general circulation of the Earths atmosphere.

Assume the equation of state given by Eq.(4.4) with ref = 1000kg m3,

= 2104 K1 and Tref = 15C, the mean temperature of the water

in the dish. The dish has a radius of 10cm and is filled to a depth of5 cm.

5. Consider the incompressible, baroclinic fluid ( = (T)) sketched inFig.8.17 in which temperature surfaces slope upward toward the poleat an angle s1. Describe the attendant zonal wind field assuming it is

in thermal wind balance.

By computing the potential energy before and after interchange of tworings of fluid (coincident with latitude circles y at height z) along asurface of slope s, show that the change in potential energy P E =P Efinal P Einitial is given by

P E = refN2 (y2 y1)

2 s (s s1) ,

where N2 = gref

zis the buoyancy frequency (see Section 4.4), ref

is the reference density of the fluid and y1, y2 are the latitudes of theinterchanged rings. You will find it useful to review Section 4.2.3.

Hence show that for a given meridional exchange distance (y2 y1):

(a) energy is released if s < s1


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8.7. PROBLEMS 285

Figure 8.17: Schematic for energetic analysis of the thermal wind considered inQ.5. Temperature surfaces have slope s1: parcels of fluid are exchanged alongsurfaces which have a slope s.

(b) the energy released is a maximum when the exchange occurs alongsurfaces inclined at half the slope of the temperature surfaces.

This is the wedge of instability discussed in Section 8.3.3 and illus-trated in Fig.8.10.

6. Discuss qualitatively, but from basic principles, why most of the Earthsdesert regions are found at latitudes of (20 30).

7. Given that the heat content of an elementary mass dm of air at tem-perature T is cpT dm (where cp is the specific heat of air at constantpressure), and that its northward velocity is v:

(a) show that the northward flux of heat crossing unit area (in thex z plane) per unit time is cpvT;

(b) hence, using the hydrostatic relationship, show that the net north-

ward heat flux H in the atmosphere, at any given latitude, can bewritten

H = cp

Zx2x1

Z

0

vT dx dz =cpg

Zx2x1

Zps0

vT dx dp ,


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Figure 8.18: The contribution by eddies ( K m s1) to the meridional flux vT(Eq.8.17), plotted as a function of latitude and pressure for (top) DJF and (bot-

tom) JJA.

where the first integral (in x) is completely around a latitude circleand ps is surface pressure.

(c) Fig.8.18 shows the contribution of eddies to the atmospheric heat

flux.What is actually shown is the contribution of eddies to the quan-tity vT, which in the above notation is

vT =1

x2 x1

Zx2x1

vT dx (8.17)

(i.e., the zonal average of vT), where x2x1 = 2a cos , where ais the Earth radius and latitude. Use the figure to estimate thenet northward heat flux by eddies across 45N. Compare this withthe requirement (from the Earths radiation budget and Fig.8.13)

that the net (atmosphere and ocean) heat transport must be about5 1015W.

chap8 gc mit

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